Understanding QFT vs QM: A Beginner's Guide to the Differences and Similarities

[Fredrik]

So I guess gauge invariance is another issue.

Yes. A gauge transformation isn't a coordinate transformation.

Are you sure spin 0 and spin 2 can't be properties of gauge invariance but only spin 1/2? How come?

No, I'm not sure because the only gauge theory I've studied is QED, and it was a long time ago. Now that you mention it, gravitons have spin 2. Not sure what that means though. There are a few threads here where the question of whether gravity is a gauge theory is debated. I think the conclusion was that it's not a gauge theory in the traditional sense, but the answer still depends on what exactly you mean by a gauge theory. I still think that what I said is correct, but if someone tells you that I'm not and they sound like they know what they're talking about, they're probably right.

Btw.. in QED.. do they analyze the electric field as coulomb potential or only as virtual particles... like every analysis in QED involves perturbation of particles?

I think it can be treated as a potential in approximate calculations, but as I said, it was a long time ago.

 

[Fredrik]

My question is. Can you make use of Gauge Theory without using Quantum Field Theory? Or the two completely related? But noether theorem can be applied to Newtonian physics so can the gauge symmetry concept of electromagnetism U(1), electroweak U(1)xSU(2), Strong SU(3) can be developed without using the concept of quantum field theory?

I didn't answer the last part. Yes, it can be developed in an entirely classical setting, using fiber bundle theory. The mathematics is pretty heavy. The classical theories that are found this way are however pretty useless until they are quantized in one way or another.

 

[atyy]

The electromagnetic field is a gauge field when potentials are used, as they are QFT.

The more common definition of a gauge field just means that several different ways of naming the field are physically equivalent. So electric potential in circuit theory has a gauge invariance in this sense - it is only potential difference that is physical, the potential itself can be shifted arbitrarily. In the same sense, the diffeomorphism invariance is a gauge invariance - metrics that are related by diffeomorphisms are physically equivalent. This is why you will see the term "de Donder gauge" with reference to classical general relativity.

There is a second different definition of a gauge field as the connection on a bundle, and gravity is not a gauge field in this sense.

 

[Fredrik]

Ah, I was confused about the most important detail. I probably shouldn't be posting this late at night. I remembered that QED is found by taking one theory and adding another field to make the theory gauge invariant. But I was thinking that this process adds the Dirac field to electromagnetism, when in fact it's the other way round. You start with the Lagrangian for a non-interacting Dirac field, note that it's not gauge invariant, and add a vector (spin-1) field with special properties to get a theory that is gauge invariant. This vector field is the electromagnetic 4-potential.

 

[atyy]

Oh yes, there's another interesting point. Actually there is more than one way to make the Dirac and EM fields interact while having EM gauge invariance. The usual "gauge principle" is maybe more informatively called "minimal coupling" - just as the "equivalence principle" of GR is really a "minimal coupling" of matter and metric.

 

[waterfall]

Ah, I was confused about the most important detail. I probably shouldn't be posting this late at night. I remembered that QED is found by taking one theory and adding another field to make the theory gauge invariant. But I was thinking that this process adds the Dirac field to electromagnetism, when in fact it's the other way round. You start with the Lagrangian for a non-interacting Dirac field, note that it's not gauge invariant, and add a vector (spin-1) field with special properties to get a theory that is gauge invariant. This vector field is the electromagnetic 4-potential.

U(1) gauge invariance is supposed to be that of electromagnetism. But in QED, electrons or spin 1/2 are involved because it is supposed to be an interaction between light and matter. So how come they sort of ignored the spin 1/2 of matter waves and just focus on the photon spin 1? Why not spin 1 + spin 1/2 which is not gauge invariant?

 

[sheaf]

Oh yes, there's another interesting point. Actually there is more than one way to make the Dirac and EM fields interact while having EM gauge invariance. The usual "gauge principle" is maybe more informatively called "minimal coupling" - just as the "equivalence principle" of GR is really a "minimal coupling" of matter and metric.

Oh that sounds interesting. How do you build the Lagrangian without resorting to minimal coupling?

 

[Demystifier]

Are you saying not all physicists with Ph.D. are experts in QFT? I thought they all wer.

That shows how little you know about physics in general. Before styding QFT, I would recommend you to start with more elementary stuff, such as classical mechanics, classical field theory, classical electrodynamics, and elementary quantum mechanics, before attempting to refute QFT theories published in peer review journals.

 

[Fredrik]

U(1) gauge invariance is supposed to be that of electromagnetism. But in QED, electrons or spin 1/2 are involved because it is supposed to be an interaction between light and matter. So how come they sort of ignored the spin 1/2 of matter waves and just focus on the photon spin 1? Why not spin 1 + spin 1/2 which is not gauge invariant?

What makes you think something has been ignored? I don't follow you here. What I said is that if you take the theory with just electrons and positrons that don't interact with anything including themselves, it's not U(1) invariant, but if you add an interaction term involving the electromagnetic 4-potential (i.e. photons), you get a U(1) invariant theory.

Oh that sounds interesting. How do you build the Lagrangian without resorting to minimal coupling?

You can add more gauge invariant terms (products of fields and field derivatives with more factors), in addition to the simplest one. The problem, in the case of QED at least, is that the simplest possibility is the only one that gives us a renormalizable theory. I think the theory with all gauge invariant terms added would be the most accurate, if we could find a way to do calculations with it, but no one cares, since the contribution from the non-renormalizable terms to low-energy processes is negligible anyway, and since the predictions made by the renormalizable theory are accurate enough that experiments with current technology can't find anything wrong with the theory.

Maybe you could also add additional gauge fields. I'm not sure. I think in that case, it wouldn't be a U(1) gauge theory anymore.

 

[sheaf]

You can add more gauge invariant terms (products of fields and field derivatives with more factors), in addition to the simplest one. The problem, in the case of QED at least, is that the simplest possibility is the only one that gives us a renormalizable theory. I think the theory with all gauge invariant terms added would be the most accurate, if we could find a way to do calculations with it, but no one cares, since the contribution from the non-renormalizable terms to low-energy processes is negligible anyway, and since the predictions made by the renormalizable theory are accurate enough that experiments with current technology can't find anything wrong with the theory.

Maybe you could also add additional gauge fields. I'm not sure. I think in that case, it wouldn't be a U(1) gauge theory anymore.

Ah thanks, I often wondered what the "minimal" in "minimal coupling" was referring to!

 

[waterfall]

In the context mentioned in this thread. How does QFT analysis differs to condensed matter physics where according to wiki: "These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system.". In normal QFT like QED, we just deal with some photons interacting with electron and you only have a few interactions in the Feynman diagrams (plus those first order perturbations or virtual particles). How about in condensed matter when there are lots of atoms. Any bird's eye view of how the analysis is done?

 

[waterfall]

What makes you think something has been ignored? I don't follow you here. What I said is that if you take the theory with just electrons and positrons that don't interact with anything including themselves, it's not U(1) invariant, but if you add an interaction term involving the electromagnetic 4-potential (i.e. photons), you get a U(1) invariant theory.


You can add more gauge invariant terms (products of fields and field derivatives with more factors), in addition to the simplest one. The problem, in the case of QED at least, is that the simplest possibility is the only one that gives us a renormalizable theory. I think the theory with all gauge invariant terms added would be the most accurate, if we could find a way to do calculations with it, but no one cares, since the contribution from the non-renormalizable terms to low-energy processes is negligible anyway, and since the predictions made by the renormalizable theory are accurate enough that experiments with current technology can't find anything wrong with the theory.

Maybe you could also add additional gauge fields. I'm not sure. I think in that case, it wouldn't be a U(1) gauge theory anymore.

Someday when we arrive at the interacting theory or know more the nature of space and time and matter, do you accept that there may be other interactions not predicted or results brought about by perturbation theory? Interactions within the dynamics of the new understanding that can for example even explain the mystery of higher temperature superconductivity, etc? Or thousands of years into the future. Do you defend that QED predictions will remain so and no new interactions even after we discover the correct interaction theory or right interpretation of quantum mechanics for example?

 

[waterfall]

It's worth pointing out that QFT is a subset of quantum mechanics. QFT is specifically the quantum mechanics of fields. So in discussions of "QM vs QFT", QM must be understood to mean "quantum mechanics of nonrelativistic point particles," and QFT must be understood to mean "quantum mechanics of relativistic fields" (one can have non-relavistic QFTs).

Fock space is a Hilbert space. QFT is just the quantum mechanics of fields, and all quantum mechanics uses Hilbert space.

This seems a bit strange; what does it mean for something to be "in physical 3D" and what does it mean for something to be "physical?"
I think you can make a strong case that at least the electromagnetic field is "physical"--it is fairly directly measurable. And the electromagnetic field, properly treated, is a quantum field.

Observables are (represented by) operators in both QM and in QFT. In QM, position is an observable; there is a position operator.

In QFT, people usually say that, in contrast to the case in QM, position is a "label" on operators. A quantum field is really a set of operators, one at each point in spacetime; i.e., an infinite set of operators, each "labelled" by a spacetime position.

I don't know what you mean by "self-observing."

As I mentioned above, observables are operators. There are momentum and angular momentum operators in QFT just as in QM, so in both cases these are observables.


You can include some effects of relativity in regular QM, but to a get a completely consistent accounting for special relativity you need relativistic QFT. QFT includes special relativity in all its aspects. All phenomena of special relativity--time dilation, length contraction, mass-energy equivalence, etc.--appear in QFT, as they must.



In QFT we define an "electron field" whose quantized oscillations are electron particles. The electron field is a bit of a weird thing, though. For instance it is not directly observable. For another its components are "Grassman numbers," as opposed to the electromagnetic field whose components are real numbers.

Just to be clear on something. Although the electron field is not observable, it has components of "grassman numbers". Now if these "grassman numbers" were altered by say the components of real numbers in the electromagnetic field, then there would be corresponding change in the electron particle even though the electron field is non-observable? Something similar to the Aharonov-Effect?

 

[waterfall]

Oh yes, there's another interesting point. Actually there is more than one way to make the Dirac and EM fields interact while having EM gauge invariance. The usual "gauge principle" is maybe more informatively called "minimal coupling" - just as the "equivalence principle" of GR is really a "minimal coupling" of matter and metric.

I think I'd better mention this here too. There is in fact more than one way to "make the Dirac and EM fields interact while having EM gauge invariance". Or in the case of Condensed Matter physics, there are other dynamics or interactions that are not commonly studied. In other words, the perturbative approach miss other interactions with significant effects. And herein may lie the answer to the puzzle of high temperature superconductivity for example.

Is anyone familiar with this or heard of this concept before?

It started in 1953 by a paper by Dicke published in peer reviewed Physical Review Journal called "Coherence in Spontaneous Radiation Processes" http://prola.aps.org/abstract/PR/v93/i1/p99_1

Then in 1973 Hepp and Lieb published in peer reviewed papers "K. Hepp, E.H. Lieb, Phys. Rev. A 8 (1973) 2517. and K. Hepp, E.H. Lieb, Ann. Phys. 76 (1973)" concerning these Super-Radiant Phase Transition in condensed matter.

Or by way of summary:

"Dicke formulated a model which was later shown to exhibit a super-radiant phase
transition (by Hepp and Lieb). The notion that such phase transitions should exist in condensed matter systems has been investigated in a series of papers by Preparata and coworkers [4–6] and others [7–9]. Different workers have come to somewhat different conclusions concerning super-radiant phase transitions [10–19]. Some doubt has been expressed [20–24] concerning the physical laboratory reality of super-radiant phase transition.

The mathematical issues are as follows: (i) It appears, at first glance, that quadratic
terms (in photon creation and annihilation operators) enter into the model via quadratic
terms inthe vector potential A. (ii) The quadratic terms in the “corrected Dicke model”
appear to destroy the super-radiant phase transition.

Then many works show that if the dipole–field interactionis treated in a gauge invariant manner [25–27] then the interaction is strictly linear in the electric field E. Thus, quadratic terms are absent for purely electric dipole–photon interactions [28]. These considerations render likely the physical reality of condensed matter super-radiant phase transitions."

(from http://arxiv.org/abs/cond-mat/0007374)

What do you make of this? Preparata and others have shown many experimental results.

http://arxiv.org/abs/cond-mat/9801248
http://arxiv.org/abs/quant-ph/9804006

Has anyone encountered the concepts mentioned in this message before? Can you please comment especially experts in Condensed Matter (and even the not so experts). If confirmed. The implications would be significant. Latest paper concerning the original peer reviewed concept or ideas was just last January 31, 2012 for example in http://arxiv.org/pdf/1108.2987.pdf

 

[muppet]

What I'm about to say is a lot more American than my taste in vernacular usually permits, but I'll say it anyway:

Dude, seriously.

You can't jump from reading popsci books to speculating on the implications of the latest research published in journals (based purely on abstracts that you, like other people who aren't specialists in the topic of the paper, don't understand), just after 64 posts of discussion on an internet forum.

If you want to understand this stuff properly, then perhaps start here. You'll need a pen, paper, coffee, and probably at least 5 years, more if you're only studying in your free time. Get back to us when you get stuck.

On the other hand, if you want to get a decent layman's understanding of what QFT is, and how it relates to the ordinary world, great. Here's a good place for that too. But keep it simple, and don't worry about what are essentially technical concepts like Fock space. It's an infinite dimensional vector space expressed as a direct sum of other infinite dimensional vector spaces. That's what it is, and if that doesn't add much to your understanding, then you're asking the wrong question for now. Far better instead to work out why you can pick up the electromagnetic field with your radio antenna, but you can't do the same for the "electron field".

 

[waterfall]

What I'm about to say is a lot more American than my taste in vernacular usually permits, but I'll say it anyway:

Dude, seriously.

You can't jump from reading popsci books to speculating on the implications of the latest research published in journals (based purely on abstracts that you, like other people who aren't specialists in the topic of the paper, don't understand), just after 64 posts of discussion on an internet forum.

If you want to understand this stuff properly, then perhaps start here. You'll need a pen, paper, coffee, and probably at least 5 years, more if you're only studying in your free time. Get back to us when you get stuck.

On the other hand, if you want to get a decent layman's understanding of what QFT is, and how it relates to the ordinary world, great. Here's a good place for that too. But keep it simple, and don't worry about what are essentially technical concepts like Fock space. It's an infinite dimensional vector space expressed as a direct sum of other infinite dimensional vector spaces. That's what it is, and if that doesn't add much to your understanding, then you're asking the wrong question for now. Far better instead to work out why you can pick up the electromagnetic field with your radio antenna, but you can't do the same for the "electron field".

I got the conceptual essentials of QFT.. now I'm reading on condensed matter physics so I understood the essence of the papers I shared above as I have the textbook about it. i want to focus on condensed matter non-relativistic QFT now as it is the heart of my interests. I want to verify or refute the Dicke theory mentioned above. Anyone want to help?

 

[muppet]

I got the conceptual essentials of QFT

Sorry, but I don't think you do.

So I guess gauge invariance is another issue. Are you sure spin 0 and spin 2 can't be properties of gauge invariance but only spin 1/2? How come?

Btw.. in QED.. do they analyze the electric field as coulomb potential or only as virtual particles... like every analysis in QED involves perturbation of particles?

Just to be clear on something. Although the electron field is not observable, it has components of "grassman numbers". Now if these "grassman numbers" were altered by say the components of real numbers in the electromagnetic field, then there would be corresponding change in the electron particle even though the electron field is non-observable? Something similar to the Aharonov-Effect?

After days of discussions. I know all of them already. But I have new questions.

I heard it said that an electron around a proton or even a traveling single electron can be modeled by QFT. So how does one start to do that? I want to imagine the matter field of electron and proton and how they behave and also the matter field of the single traveling electron. I know QFT is appropriate for an "infinite numbers of particles". But again I heard it can be done for a single or two particles. How?

My question is. Can you make use of Gauge Theory without using Quantum Field Theory? Or the two completely related? But noether theorem can be applied to Newtonian physics so can the gauge symmetry concept of electromagnetism U(1), electroweak U(1)xSU(2), Strong SU(3) can be developed without using the concept of quantum field theory?

 

[DarMM]

If Fock space is the Hilbert space for non interacting quantum fields, then what is the corresponding space for interacting quantum fields? And what is it supposed to mean the quantum field is not interacting?

The answer to this is very boring, but it's just some other Hilbert Space. A Fock space is a Hilbert space which is the sum of direct products of some other smaller Hilbert Space, for the interacting Hilbert space this is not true.

Also, for example in \phi^{4}_{2}, the theory requires a different Hilbert space for each value of the coupling \lambda as proven by Nelson.